18 Jul 2013

Sum of Diagonals: Upper Secondary Mathematics Competition Question

The figure ABC is a right-angled triangle with lengths AB = AC = 4. Point A1 is the mid-point of the line AC; the point A2 is the mid-point of the line segment A1C, and so on. The line segment A1B1 is perpendicular to AC and intersects the line BC at point B1; similarly A2B2 is perpendicular to AC and intersects BC at B2, and so on.

This construction gives us the diagonal line segments of lengths BA1, B1A2, B2A3 and so on. Calculate the sum of the infinite series

D = BA1 + B1A2 + B2A3 + B3A4 + ...

Hence, if H is the length BC, find the value of D/H.

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